What is a p-value?

Anyone who has taken a basic statistics course remembers the term “p-value” and that when it’s a certain value you reject something and when it’s not you do something else. Who knows, I can’t remember. (Kidding, I teach statistics!)

Well, simply put, a p-value is the probability of getting the data you have if you assume something beforehand. Put mathematically, this is $P(\text{data}|\theta)$, where $\theta$ is some parameter you are interested in, like the mean. So let’s say we are interested in the average weight of all males at a certain company. We start by assuming it’s 185 lbs and conduct a two sided hypothesis test. Our p-value is 0.022 so we reject the assumption that the mean weight is 185. What we have actually found is $P(\text{data}|\theta=185)=0.022$. So if the real mean weight is 185 lbs, when we randomly sample a group we will get the results of our data or more extreme 2.2% of the time. Seems useful right? Maybe, maybe not. All we know is that is seems kind of unlikely to happen if the real mean weight were 185 lbs. Seems, maybe, unlikely? Ugg. Statistics deals with uncertainty but this level of uncertainty doesn’t cut it for me.

This kind of hypothesis testing tells us NOTHING about the range of the mean weight. What if we are interesting in $P(\theta=185)$? Or what if we want to know a range like $P(180 <\theta<190)$? That would be very powerful information! This probability can not be found through traditional hypothesis testing and requires uses Bayesian techniques. It has become a problem that studies find a “significant p-value” of something less than 0.05 and claim to have significant results that go against everything known about that field. Recently the American Statistical Association (AMA) commented on this problem and how this way of thinking needs to go. Hypothesis testing is a valid tool but it is much less powerful than it seems at first glance and is too often completely misunderstood.

In the war between Frequentists and Bayesian followers, it seems the AMA is siding with the latter group in this case.

What is the ambiguous case?

In high school geometry, the idea of proofs are often first introduced to American students. A common task is to use a basic set of rules to prove two triangles are congruent, which is a fancy way of saying they have the same side lengths and the same angles. Phrases like side-angle-side (SAS) and angle-angle-side (AAS) might ring a bell. This topic is very polarizing and quickly separates students into those who say “I love geometry!” and “I miss algebra!”. Simply put though, all of these acronyms are just descriptions of the information we are given about a triangle. Note that the order of this information is very important! SAS and SSA both contain one angle and two sides, but both of them won’t prove two triangles are congruent.

There is one problematic case of side-side-angle (SSA) or put another way, angle-side-side (I’ll pause here and let you chuckle at the joke every geometry teacher in the country makes at this moment). Some teachers will correctly note that this situation is not enough to show two triangles are congruent and leave it at that, but there is more that can be explored here. When you are given a SSA triangle, this is called the ambiguous case because it could in fact result in 0, 1, or 2 triangles. This post will focus on demonstrating that no solutions exist.

Step-by-step method of solving

Draw the triangle and label all given information. It doesn’t need to be to scale at all! It also doesn’t matter how you orient the triangle as long as side a is across from angle A, side b is across from angle B, and side c is across from angle C. Just draw and label.

Use the Law of Sines to solve for the angle across from the second side. Which one is the second side? Well in SSA you have two sides. One of the angle will pair with a side, like angle A and side a, or angle B and side b. There will be an unused side that isn’t paired. We are going to try to find that angle.

Let’s walk through an example.

Side-Side-Angle Triangle Example$B = 55 ^{\circ}, b=8.99,a=26.22$

Following the steps above, let’s draw this triangle!

Notice that angle A is in the bottom right corner. Often students and teachers like to put angle A at the top. It doesn’t matter as long as the sides go across from the angles! You could draw the triangle like this as well. It won’t change the answer.

The Law of Sines states that $\displaystyle \frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}$ As mentioned in step 2, we are going to try to find angle B, because it is the angle opposite the second side. (We can’t find angle C or side c right now because we don’t have any information on that side. Try to solve for either of those to convince yourself if that seems unclear.)

Uh oh! We have a problem!! Remember that the range of $\sin(x)$ is $[-1,1]$, meaning the output of this function has to stay between these two values. When we get an answer like the one above, it is IMPOSSIBLE to find an angle to make the statement true and we have an IMPOSSIBLE triangle. Try putting $\sin^{-1}(2.389)$ into your calculator to check and you’ll get an error message.

Key Points

Draw a triangle and label your given information. Do not worry at all if the triangle is to scale. Use the Law of Sines to find the second angle that connects to the second side you are given and simplify. When you get $\sin(A), \sin(B), \sin(C)$ is outside of $[-1,1]$, you can confidently answer that this triangle does not exist. If it does fall inside that range, then you should move onto deciding if 1 or 2 triangles are possible. That topic will be covered in another blog post soon.

Sir William Timothy Gowers is, as you might expect, a British mathematician. As with Aaronson, I won’t repeat his biography here, as the wiki is adequate. I just thought I’d point out one fun little tidbit: Gowers once sang in the King’s College Choir, Cambridge, as a choirboy. That choir is one of the best in the world, and is probably my favorite. In particular, their album O Come All Ye Faithful is one of the truly great Christmas albums.

Gowers’s style is engaging, while still technical enough to interest the expert. I have linked to several posts of interest. I just thought I would especially point out the Polymath5 tag, which corresponds to the Erdos Discrepancy Problem. This is an attempt to solve a mathematical problem with an highly collaborative effort. Probably it is not the case that any ol’ problem can be solved this way, but there are features of certain kinds of problems which render them vulnerable to this kind of attack. It’s an interesting experiment. Another interesting experiment was Gowers’s proof program for generating proofs of certain kinds of theorems. Yours truly submitted one comment for that experiment.

One criticism I might have is that Gowers does not usually explicitly mention the prerequisites necessary to understand various posts. Aaronson doesn’t do this, either. This unfortunately makes the signal-to-noise ratio lower than it might be. If I have no idea how to start reading a post because it dives into technical details way over my head, then I’m less likely to come back for more. Naturally, it’s impossible to please everyone, and some might find a list of prerequisites for each post tiresome. On blogs, at least, you could use tags to accomplish this. Then it’s there, but not in-your-face.

In conclusion, I would definitely say this blog is worth following, even if its frequency is a bit low. The blog is still alive certainly; posts are dense and efficient, as you’d expect from a Fields-medal-level mathematician.

Shtetl-Optimized, as I’ve mentioned elsewhere, is the blog of Scott Aaronson. And, as I’ve just linked to his wiki page, I don’t feel especially inclined to say much about his biography. Aaronson’s field of research is quantum computation; an exciting new field with possibilities. It lies at the intersection of mathematics, physics, computer science, and electrical engineering.

I have found Shtetl-Optimized to be well-written and informative. His latest post, dated March 22nd, is about Tegmark, his book Our Mathematical Universe, and Tegmark’s idea of the Mathematical Universe Hypothesis. Aaronson’s writing is lively and witty, you’ll find. He writes about both the human-ness of Tegmark’s writing, as well as his disagreement with Tegmark’s fundamental thesis, which is that the universe is not just mathematical, but is itself the mathematics. One of Aaronson’s reasons for finding the Mathematical Universe Hypothesis (MUH) unconvincing is that it is not impressive science – defined as when elegant mathematics lines up with facts as we have experienced them. I think this is exactly right. It does seem unlikely, as well, that anyone could ever devise experiments to test the MUH. This is my issue with all multiverse theories: while theories are definitely a part of science, and important (theories are the goal, after all!), experiment is the life-blood. A theory without experiment exists only in men’s minds, and has dubious predictive power.

Shtetl-Optimized is probably not for the layman, however, as he has no hesitation in getting “technical”. If you are interested in popular science, this blog would stretch you. Perhaps that is what you’re seeking, and if so, you’ve found a good place. It’s like classical music: you get out of this blog what you put into it.

It seems to me that great mathematical writing is rare, and to be celebrated. It follows that pointing out great mathematical writing, as well as poor mathematical writing, could be a very useful function of this blog. So I propose this as one feature of the blog.

We can review books, blogs, articles, etc. Any mathematics in print is fair game. What should be the criteria by which we judge mathematics to be well-written or not? To some extent, the rules of basic English should apply. We should see punctuated equations, as per N. David Mermin, correct grammar and syntax, and consistent formatting. In addition to these low-level necessities, we should see careful definitions, a concern for the reader, a lively, interesting, engaging style, as well as clarity of expression.

One aspect of mathematical writing not often brought to the fore is the difference between research and scholarship, as mentioned in Morris Kline’s book Why the Professor Can’t Teach, to which I linked above. Research is coming up with new mathematical theorems, procedures, etc. Scholarship is organizing, codifying, and clarifying existing research. One quote from Kline’s book (which I quote loosely) is that “One good scholarly paper is worth a hundred research papers.” Having attempted to read a number of research papers, I can definitely say that the vast majority of them are exceptionally poorly written, tending to be esoteric for the sake of being esoteric, and are generally useless except for the ultra-specialist.

It was V. I. Arnold who wrote the following:

It is almost impossible for me to read contemporary mathematicians who, instead of saying “Petya washed his hands,” write simply: There is a $t_1 <0$ such that the image of $t_1$ under the natural mapping $t_1 \mapsto \x{Petya}(t_1)$ belongs the set of dirty hands, and a $t_{2}, t_{1}<t_{2}\le 0,$ such that the image of $t_2$ under the above-mentioned mapping belongs to the complement of the set defined in the preceding sentence.

This is exactly right. It is this sort of obfuscated “mathematicalese” that I would combat, and I would welcome fellow fighters in this regard.

Testing the math delimiters: $$\int_{-\infty}^{\infty}e^{-x^2} \, dx= \sqrt{\pi}.$$ Success! We appear to have MathJax on the blog now. This will obviously greatly facilitate the posting of tutorials, or any discussion at all concerning math, as $\LaTeX$ is (mostly) available.

Testing: $A \; \x{implies} \; B$. This means that my new command of \x{} is equivalent to \text{}, and works just fine.

Welcome to Math Help Blogs! As you can see in the subtitle, this blog is a companion to the Math Help Boards, an up-and-coming free math help website with the following unique characteristics:

1. Warm and welcoming atmosphere.

2. Distributed authority, with delegation. We have four different staff ranks, with great fluidity between ranks. We’ve had one user climb up all the way up from regular user to system administrator!

3. Completely spam-free, due to #2 above, as well as well-chosen anti-spam software such as Spam-O-Matic

4. Good teaching philosophy: we do not simply hand out the answers. We ask users to put forth effort and show us their work so far. Then we get them unstuck. This maximizes the teaching moment, and helps the student do the heavy lifting.

5. Highly active administration. There are four regular administrators, all of them active helping to improve the site. We are two years old, and with this level of sysadmin, you can bet we’ll be around for a long time!

With these factors in place, we have experienced steady growth from the get-go. We now have over 1600 users, almost 200 of which are active (have logged in in the past month), almost 10k threads, with over 44k posts. While these statistics may seem small compared with sites like the excellent Physics Forums, we are only two years old.

So much for Math Help Boards. What about Math Help Blogs? This blog exists to highlight events at Math Help Boards, as well as talk about mathematics and mathematics education in general, and possibly other topics as well. We’ll define ourselves as we go.

In the meantime, welcome to Math Help Blogs, and don’t neglect to check out Math Help Boards!